Test: Absolute Values/Modules - GMAT MCQ

Statement (1): x > 10

This statement tells us that x is greater than 10. However, it does not provide any specific information about x's relationship with 30. Therefore, it is not sufficient to answer the question.

Statement (2): x > 25

This statement tells us that x is greater than 25. Similarly to statement (1), it does not provide any specific information about x's relationship with 30. Therefore, it is also not sufficient to answer the question.

When we consider each statement alone:

Statement (1) alone does not provide enough information to determine the relationship between |x - 10| and |x - 30|.
Statement (2) alone does not provide enough information to determine the relationship between |x - 10| and |x - 30|.

When we consider both statements together:

By combining the two statements, we have x > 10 and x > 25. Since x has to be greater than both 10 and 25, we can conclude that x > 25.

Now, let's compare |x - 10| and |x - 30|:

When x > 25, we can simplify |x - 10| and |x - 30| as follows:

|x - 10| = x - 10 (since x - 10 is positive when x > 10)
|x - 30| = x - 30 (since x - 30 is positive when x > 30)

Now we can rewrite the original inequality:

x - 10 > x - 30

By subtracting x from both sides of the inequality, we get:

-10 > -30

This statement is true. Therefore, |x - 10| is greater than |x - 30|.

Hence, when considered together, statement (2) alone is sufficient to answer the question asked. The answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): p is prime

This statement tells us that p is a prime number. A prime number is defined as a number greater than 1 that has no positive divisors other than 1 and itself. Since p is a prime number, it is always positive. Therefore, we can conclude that p = |p|. Statement (1) alone is sufficient to answer the question.

Statement (2): p is even

This statement tells us that p is an even number. However, it does not provide information about the sign of p. Even numbers can be positive or negative. Therefore, we cannot determine whether p = |p| based on statement (2) alone. Statement (2) is not sufficient to answer the question.

When we consider each statement alone:

Statement (1) alone is sufficient to determine that p = |p|.
Statement (2) alone is not sufficient to determine whether p = |p|.

When we consider both statements together:

Even though statement (2) tells us that p is even, it does not provide information about whether p is positive or negative. Therefore, combining the statements does not provide enough information to determine whether p = |p|. Statements (1) and (2) together are not sufficient to answer the question.

Hence, the answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): |x| > |y|

This statement tells us that the absolute value of x is greater than the absolute value of y. However, it does not provide any direct information about the signs of x and y. Therefore, we cannot determine the relationship between x and y based on statement (1) alone.

Statement (2): x = |x|

This statement tells us that x is equal to the absolute value of x. From this statement, we can conclude that x is either a non-negative number (x ≥ 0) or zero (x = 0). However, it does not provide any information about y or the relationship between x and y. Therefore, we cannot determine the relationship between x and y based on statement (2) alone.

When we consider each statement alone:

Statement (1) alone is not sufficient to determine the relationship between x and y.
Statement (2) alone is not sufficient to determine the relationship between x and y.

When we consider both statements together:

Combining the statements, we know that |x| > |y| and x = |x|. From this, we can conclude that x is a non-negative number (x ≥ 0). However, we still do not have enough information to determine the relationship between x and y. For example, if x = 0 and y = -1, then x is not greater than y. On the other hand, if x = 1 and y = -1, then x is greater than y. Therefore, statements (1) and (2) together are not sufficient to answer the question.

Hence, the answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): |x| = 4

This statement tells us that the absolute value of x is 4. However, it does not provide any direct information about the sign of x or y. Therefore, we cannot determine the value of |x + y| based on statement (1) alone.

Statement (2): |y| = 1

This statement tells us that the absolute value of y is 1. Similar to statement (1), it does not provide any direct information about the sign of x or y. Therefore, we cannot determine the value of |x + y| based on statement (2) alone.

When we consider both statements together:

Combining the statements, we know that |x| = 4 and |y| = 1. From this, we can conclude that x could be either 4 or -4, and y could be either 1 or -1. Since xy < 0, we can deduce that x and y have opposite signs.

To find the value of |x + y|, we need to consider all possible combinations of x and y with opposite signs.

If x = 4 and y = -1, then |x + y| = |4 + (-1)| = |3| = 3.
If x = -4 and y = 1, then |x + y| = |-4 + 1| = |-3| = 3.

Therefore, regardless of the specific values of x and y, the value of |x + y| is always 3.

Hence, the answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): |A| = -|B|

This statement states that the absolute value of A is equal to the negation of the absolute value of B. Since absolute values are always non-negative, the only way for |A| to be equal to -|B| is if both A and B are equal to 0. In that case, the sum of A and B is 0.

Statement (2): |B| = -|A|

This statement states that the absolute value of B is equal to the negation of the absolute value of A. Similar to statement (1), this implies that both A and B are equal to 0, resulting in a sum of 0.

From both statements, we can conclude that the sum of A and B is always 0.

Therefore, each statement alone is sufficient to answer the question asked. The answer is (D).

Statement (1): |x+2| ≤ 4

This statement tells us that the absolute value of x+2 is less than or equal to 4. It means that the distance between x+2 and 0 on the number line is less than or equal to 4. This can be represented as -4 ≤ x+2 ≤ 4. By solving this inequality, we find that -6 ≤ x ≤ 2. Therefore, the possible values of x lie between -6 and 2, inclusive.

Statement (2): x² = 36

This statement tells us that the square of x is equal to 36. Taking the square root of both sides gives us two possible solutions: x = 6 and x = -6.

When we consider both statements together, we find that the possible values of x are -6, -5, -4, -3, -2, -1, 0, 1, 2. Therefore, the value of x cannot be determined uniquely based on the given statements.

Hence, both statements (1) and (2) together are sufficient to answer the question asked, but neither statement alone is sufficient. The answer is (C).

Statement (1): m < 0

This statement provides a condition on the value of m, but it does not give us enough information to determine the specific value of m. It does not directly help us solve the equation |m + 4| = 2.

Statement (2): m² + 8m + 12 = 0

This statement gives us a quadratic equation involving m. However, it is not directly related to the absolute value equation |m + 4| = 2. Solving this quadratic equation may give us the possible values of m, but it does not guarantee that these values satisfy the given absolute value equation.

Considering the statements individually, neither statement alone is sufficient to determine the value of m.

If we consider both statements together, we still cannot determine the value of m. Although statement (2) provides a quadratic equation, it does not directly relate to the absolute value equation in statement (1).

Therefore, together, statements (1) and (2) are not sufficient to answer the question, and additional data is needed.

The correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1): |x - 2| = 1

This statement tells us that the absolute value of the difference between x and 2 is equal to 1. This can be rewritten as two separate equations: x - 2 = 1 and x - 2 = -1. Solving these equations individually gives us x = 3 and x = 1. Therefore, statement (1) alone provides two possible values for x: x = 3 and x = 1.

Statement (2): x² = 4x - 3

This is a quadratic equation that can be rewritten as x² - 4x + 3 = 0. Factoring or using the quadratic formula, we find that the solutions to this equation are x = 1 and x = 3. Therefore, statement (2) alone also provides two possible values for x: x = 1 and x = 3.

When we consider both statements together, we find that the only common solution to both equations is x = 3. Therefore, the value of x is 3.

However, since both statements provide multiple possible values for x individually, and the value of x is not uniquely determined by the given information, we cannot definitively answer the question with certainty.

Therefore, the correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1): The product of the two numbers is 6.

This statement tells us about the product of the two numbers but does not provide direct information about their sum or their individual values. It does not directly help us determine the absolute value of their sum.

Statement (2): One number is 5 less than the other number.

This statement gives us a relationship between the two numbers, but it still does not provide specific values or direct information about their sum. It does not allow us to determine the absolute value of their sum.

Considering the statements individually, neither statement alone is sufficient to determine the absolute value of the sum of the two numbers.

However, if we consider both statements together, we can find the values of the two numbers. Since the product of the two numbers is 6, the possible pairs of numbers are (1, 6) and (-1, -6). From statement (2), we know that one number is 5 less than the other. The only valid pair that satisfies this condition is (1, 6), where 6 is 5 more than 1. Thus, the sum of the two numbers is 1 + 6 = 7, and the absolute value of their sum is |7| = 7.

Therefore, together, statements (1) and (2) are sufficient to answer the question, but neither statement alone is sufficient.

The correct answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): x - y = 6

This statement provides information about the difference between x and y but does not directly relate to their absolute values. It does not provide enough information to determine if |x| = |y|.

Statement (2): x + y = 0

This statement tells us that x and y are additive inverses of each other, meaning they have equal magnitudes but opposite signs. Since the absolute value of a number represents its magnitude, we can conclude that |x| = |y|.

Considering the statements individually, only statement (2) provides sufficient information to determine that |x| = |y|. Therefore, statement (2) alone is sufficient to answer the question, but statement (1) alone is not sufficient.

Hence, the answer is (B) statement (2) alone is sufficient, but statement (1) alone is not sufficient to answer the question asked.